If $A$ is a nonsingular matrix, then $A=E_n...E_2.E_1$, where $E_n$ are elementary matrices

How do I prove that

If $A$ is a nonsingular matrix, then there exists elementary matrices $E_1,E_2,E_3....E_n$ such that, $$A=E_n...E_3.E_2.E_1.I=E_n...E_3.E_2.E_1$$

My Understanding:

I feel this got to be true from the row operations that we use to find $A^{-1}$ from the equation $A=IA$, but how do I prove it mathematically ?

Remember that every elementary operation on the rows of $\;A\;$ is a product $\;EA\;$ ,where $\;E\;$ is an elementary matrix. Observe $\;E\;$ multiplies from the left, otherwise that'd be an elementary operation on the columns of $\;A\;$ .
• @ss1729 Exactly as shown in the hint...! First elem. op.: $\;E_1A\;$, second elem. op.: $\;E_2E_1A\;$ , etc. You must also know/remember that every elementary matrix is invertible...that's all! Feb 15, 2018 at 9:19
• $A^{-1}=IA^{-1}\implies E_1A^{-1}=E_1IA^{-1}\implies E_2E_1A^{-1}=E_2E_1IA^{-1}\\\implies I=E_n..E_2E_1IA^{-1}\implies A=E_n..E_2E_1I=E_n..E_2E_1$. Is it a well defined proof ?. I am truly feel thr s something missing here. Feb 15, 2018 at 19:43